Greenhouse Effect, Limited
This particular instance of radiation absorption is not physically different from the light absorption used by spectroscopic thecniques to measure the concentration of a chemical compound in a given solution. Absorption is described by Beer-Lambert's law, which has the form:
log(I/I0) = -c[J]l
or, as exponential:
I/I0 = 10-c[J]l
Above, I is the intensity of transmitted light and I0 the intensity of incident light; c a proportionality constant specific for each analyte; [J] the molar concentration of the analyte and l the length of the optical path. An example plot, using c=0.5 and l=1 can be seen here too. As explained in the text, the concentration increase from 0 to 0.5 (mol/L) produces a bigger transmittance reduction than concentration going from 10 to 20.
Going back to the greenhouse effect, what happens in case of doubling atmospheric CO2 concentration depends from in which region the doubling occurs: if it is at low concentration, the effect will be great; but near the saturation point, it will be small.
I expect climate models to take this simple issue into account - though it is a bit difficult to know where exactly is the current status of atmosphere regarding transmittance. But this is a problem that only a few laypeople know, while it is important in the AGW debate.
Update 20/02: I don't know why I forgot to include a CO2 absorption spectrum. So, here they are. The figure is carbon dioxide's absorption spectrum in laboratory conditions, while here is the actual IR emission spectrum of Earth.